If you want to do ‘best practice’ then change permissions to a higher number to get your installation issue resolved, then immediately change them back to as low a number as possible using those numbers I mentioned in the article as your starting point. Yes, you should do this for sub-folders too.
Security-wise, leaving permissions set too high will leave an open door for an attacker, so make sure these are locked down as my suggestions.

I tried deleting the “Upgrade” folder, but that didn’t work. What did work was going into my CPanel for the website and doing the upgrade with Fantastico. The WordPress was originally installed through that, and it had been years since an update had been done (not my department! LOL). Even though Fantastico was one or two upgrades behind, once it had done it’s upgrade thing, the WordPress updates could be done on their own, on the program, the themes, and the plugins. All is well now, thanks for the great ideas everyone.